let us assume the needle falls at an angle @ with the parallel lines
so the band or area it can be said to cover along 1 line= 2l|sin@| now for no intersection, this band should lie within the lines so P=2l|sin@|/2a =l|sin@|/a
now @ moves from 0 to 2 pi continuously so P=integral of(lsin@/a) from 0 to 2 pi note the mod sign
so P=4l/a edit
divide by the sample space(which is from 0 to 2 pi) P=4l/a*1/2 pi =2l/a pi
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